DESCRIPTION (taken from the applicant's abstract): Measuring in vivo the input of biological/physiological system, e.g. the time course of the secretion rate of a gland or the time course of the production rate of an endogenous substrate, is essential to the understanding of its function both in healthy and disease states. Unfortunately these inputs are usually not directly measurable in vivo. One is only able to measure the causally related effects of these inputs in the circulation, usually the time course of the concentration of the substance in plasma. There is thus the need to reconstruct the unknown causes (e.g. hormone secretion rate) from the measured effects (e.g. hormone plasma concentration). Thus the input estimation problem is an inverse problem which is usually attacked by deconvolution. However, deconvolution is well-known in the literature to be difficult since it is ill-conditioned, i.e. a small percent error in the data results in a much greater percent error in the estimated input. In addition dealing with physiological signals add to the complexity of the problem, e.g. they are often sampled at a nonuniform and infrequent sampling rate. The specific aims described in this application are to develop the theory, the numerical algorithms and a software tool to solve the input estimation problem for biological/ physiological systems by deconvolution. A novel approach is proposed to attack the ill-conditioning of the problem based on a regularization method set in a stochastic embedding and on linear minimum variance estimation. The approach is nonparametric, i.e. it does not postulate any functional form for the unknown input. The theory, also by virtue of the stochastic context, is able to cope with a number of issues which are particularly important when dealing with physiological systems, e.g. choice of the regularization parameter, case of nonuniform and infrequent sampling, estimation of the confidence intervals of the reconstructed input profile also accounting for model uncertainty; case of time-varying impulse response of the system, and nonnegative constraint on the input signal. Most of this theory has been developed but a few issues still require further research. New numerical algorithms are proposed to compute efficiently the solution of the deconvolution problem. The numerical aspects issue is mandatory since theory can be useless if there are no reasonably efficient tools to implement it in practice. In this research proposal an original singular value deconvolution strategy is proposed which dramatically speeds up the determination of the input estimate, thus permitting the solution of high-dimensional problems using low-cost hardware. Finally, a software tool is proposed usable by the physiological/clinical investigator for solving real world deconvolution problems. The software will compute the input estimate with high resolution, much finer than the experimental sampling grid, together with information on the reliability of the reconstructed input, e.g. its confidence limits. The software will be designed to deal with both standard and nonstandard problems.